Introduction RIEMANN SURFACES AND THE THETA FUNCTION

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RIEMANN SURFACES AND THE THETA FUNCTION

BY

J O S E P H L E W I T T E S

Yeshiva University, New York, N.Y., U.S.A. (1)

Introduction

The purpose of this p a p e r is to present a clear exposition of certain theorems of Riemarm, n o t a b l y Theorem 8 below, which he obtained from his s t u d y of the 0 function as a means of solving the Jacobi inversion problem. A perusal of R i e m a n n ' s collected works, [4], shows t h a t this was a topic of great interest to him. F o r this paper, one m a y consult [4], pp. 133-142, 212-224, 487-504, and the Supplement, pp.

1-59. Many mathematicians, in the half century after Riemann, tried to elucidate and justify his results. I n this connection we m a y mention Christoffel, Noether, Weber, Rost, and Poincard. Citations of the older literature m a y be found in the books, [2], [3], and [6].

Despite all these efforts, it is difficult for me to say whether or not complete proofs have been given to everything t h a t has been claimed. I n this paper, we hope to give correct proofs of some of these interesting results, along with some new theorems. Our method is essentially t h a t of R i e m a n n a n d his followers, although the language m a y be slightly more modern. The k e y to our method is consideration of the role of the base point, i.e., lower limit of the integrals of first kind, and its in- fluence on the vector K of R i e m a n n constants. The roles of the base point and K seem to have been overlooked b y all, p r o b a b l y because of the s t a t e m e n t of Riemann, [4], p. 133 and p. 213, that, under a suitable normalization, the vector K vanishes.

Finally, having available the concept of an a b s t r a c t Riemann surface gives one a distinct advantage over being tied down to a particular branched covering of the sphere.

I n the first section, we prove the basic theorem concerning the zeros of certain

"multiplicative functions". On the whole, in this section, we t r y to conform with the (1) Supported by N.S.F. G 18929. The author wishes to express his thanks to Professor H. E.

Rauch for his valuable advice and encouragement in the preparation of this paper.

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38 J. LEWrrTES

notation of Conforto's book, [1], whose chapter on t h e t a functions was v e r y helpful.

I n the second section, we study the identical vanishing of the 0 function and prove T h e o r e m 8, the main theorem of Riemann. The third section contains a n u m b e r of miscellaneous applications, and we conclude with a discussion of the hypereUiptic case, m o t i v a t e d b y Riemann's remarks in the Supplement of [4], pp. 35-39.

We assume known most of the standard function theory on R i e m a n n surfaces;

the R i c m a n n - R o c h theorem, Abel's theorem, the Weierstrass gap theorem, the properties of Weierstrass points, and the structure of hyperelliptic surfaces. We write divisors multiplicatively and use the R i e m a n n - R o e h theorem as follows: F o r a n y divisor $, the dimension of the complex vector space of meromorphie functions on the surface which are multiples of t -1, r(t-1), is given b y d e g r e e ( t ) + i ( t ) + 1 - g . Here, degree(t) is the sum of the exponents of ~, i(~) the dimension of the space of abelian differentials which are multiples of ~, and g the genus of the surface. I n particular, we use a consequence of the R i e m a n n - R o c h theorem, t h a t given a n y point P on the surface S of genus g, one m a y choose a basis for the g dimensional space of differentials of the first kind ~1 . . . ~ , such t h a t ~ has at P a zero of order n~ - 1 , where n 1 .. . . . ng are g gaps at P. F o r details one m a y consult t h e book b y Springer, [5; Chapter 10].

C o denotes the space of g complex variables. A point u E C o is considered a column vector, i.e. a g b y 1 matrix. I f A is a matrix, ~ denotes the transpose of A.

While preparing this paper, I was informed b y Prof. D. C. Spencer t h a t Theorem 8 below, R i e m a n n ' s theorem on the vanishing of the 0 function, was proved b y A. Mayer in his Princeton thesis. U p o n completion of this paper, I sent a copy to Prof, D.

Mumford who communicated to me t h a t he h a d found a proof of the a b s t r a c t algebro- geometric formulation of R i e m a n n ' s theorem, which is valid for a r b i t r a r y characteristic, not only characteristic zero.

Section I.

L e t S be a compact R i e m a n n surface of genus g/> 2 and aj, bj, 1 <~ ~<~g, the 2g cycles of a canonical dissection of S. These cycles are 2g closed curves on S which begin and end a t a common point and have the following intersection numbers:

aj•215 as•

the Kronecker delta. W h e n S is cut along these cycles, one obtains a simply connected region S o with oriented b o u n d a r y ~S0, traversed in the positive direction as a l b l a [ 1 b~ 1 ... ao bga~lb~ 1. The homology classes of these 2g cycles generate the first

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R I E M A N N S U R F A C E S A N D T H E T H E T A F U N C T I O N 39 homology group of S. Corresponding to these 2g cycles, there is uniquely determined a basis, ~1, .-.,~g, of the complex g dimensional space of abelian differentials of the first kind on S, b y the normalization,

S~,q~j=6sk~i, (i= ~

1). Setting

tjk= Sb, q;j,

one obtains the

g•

period matrix ~ =

(zdI, T),

where I is the

g•

identity matrix and T = (tjg) is a non-singular symmetric matrix with negative definite real part. I t can be shown t h a t the 2g columns of the period matrix are independent over the reals and generate a discrete abelian subgroup A of C a. The Jacobian variety, J(S), is the quotient group

Cg/A,

a compact abelian group. If u 1, u 2 are two points in C g, then we write u l ~ - u 2 if they are congruent modulo A. Thus, u 1 - u ~ if and only if u 1 - u 2 is a linear combination with integer coefficients of the columns of the period matrix, i.e., u 1 - u ~= ~ m , where m is a 2gx 1 vector of integers.

The significance of

J(S)

is t h a t there is a map S--> J(S), defined as follows. Fix a n y point B 0 G S as base point and for each point P E S choose a p a t h from B 0 to P.

Set

^uj(P)= F

~0j, 1 </~<g,

o

where the integral is taken along the chosen p a t h and denote by

u(P)E C g

the vector

(ul(P) .... ,ug(P)).

For another choice of p a t h one m a y obtain a different vector u(P), but it is clear t h a t all values of u(P) are congruent modulo A, hence determine in a well-defined manner an element of

J(S).

This gives then a map of S into

J(S).

For convenience we shall denote this map simply b y

P--> u(P),

where it is understood that

u(P)

is a n y representative in C g of the point of

J(S)

into which P is mapped.

~t

Of course, the map

S--->J(S)

depends in a vital way upon the choice of the point B 0 and this will be discussed later. T h e map u m a y be extended to map the group of divisors of S into

J(S)

b y defining for a n y divisor $ = p ~ l . . , p~k,

k

u(~) = ~ nju(Pj).

t = 1

The

degree

of ~ is the sum of the exponents;

k

deg (~)= ~ n j .

1=1

One says t h a t two divisors are

equivalent,

$1 "~ ~2, if the quotient ~-1 $~1 is the divisor of a function. Abel's theorem states t h a t

~1 N ~2 if and only if deg ($1)=deg (C2) and u(~l)-=u(~2).

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4 0 J . L E W I T T E S

Since there is no function on S with only one pole, u ( P ) ~ u ( Q ) i f P # Q are two

u

points of S, so t h a t S.--->J(S) is a one-one imbedding of S in J(S). I t is also non- singular, for the differential of the mapping at P E S is simply du = (d% (P) ... dug (P)), which has maximal rank, i.e. one; for, duj (P) i s - - i n a suitable local p a r a m e t e r - - o n l y qj (P), and it is a well-known consequence of the R i e m a n n - R o c h theorem t h a t not all differentials of the first kind vanish a t P.

Let ek be the kth column of the 2g by 2g identity matrix; then ~ ek is the kth column of the period matrix. We wish to consider functions l(u), holomorphic in all of C g, with the following periodicity property:

/(u + a eL) = exp [2.iek (Au + y)]/(u), ⁽¹⁾ where 1 ~< k ~< 2g, A, y are matrices of complex numbers, of respective size g by 29 and 2g b y 1. The equality ](u+~e~,+~eh)=[(u+~eh+~ek), implies (cf. [1], p. 57) the relation

~A-X~=~v, (2)

where N is a 2 9 by 2 9 skew symmetric matrix of integers. N is called the characteristic matrix of t.

Such functions are n o t well defined on 3(S) b u t are "multiplicative functions"

there. Nevertheless, since the multipliers are exponentials which can never vanish, it is clear that if u 1 = u 2, then [(u 1) = 0 if and only if [(u ~) = O. Hence, one m a y say in a meaningful way t h a t a point of J(S) is, or is not, a zero of f. B y means of the map u : S ~ J(S), f(u(P)) is a multiplicative holomorphie function on S. I n particular, choosing a definite value for u(Bo), which is, of course, always ~ 0, ](u(P)) is a single valued holomorphic function in the simply connected region S o with well-defined values on aS 0. When continued over ~S0, a new single valued branch of [(u(P)) on S 0 is obtained, which has the same zeros as the first branch. Whenever we write l(u(P)), we assume some such single valued branch chosen, b u t which particular one is irrele- v a n t to our present purpose.

I t is possible t h a t [(u(P))-~O on S. Here we use " ~ 0 " to mean t h a t the function is identically zero for all P, n o t to be confused with " ~ " meaning congruence modulo period vectors. I n a n y given context only one meaning will be possible. Now, if ](u(P))~0, then, b y the compactness of S, it has only a finite number of zeros on S. We m a y assume t h a t these do not lie on ~S0, for the canonical cycles m a y be deformed slightly into homologous ones, without affecting a n y of the canonical

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R I E M A N I ~ S U R F A C E S A N D T H E T H E T A F U N C T I O N

properties or the period matrix.

her, N(/), of zeros of / on S.

41 L e t us determine b y the residue theorem the num-

~(/) = ~ i ~.-]-" (3)

L e t /~ denote the value t a k e n b y / a t a point of ~S o lying on ak or b~, while / - denotes the value t a k e n a t the identified point on a~ 1 or b; ~ respectively. We also write u +, u - with the same meaning. Then

2~i k = l ~,+ k

We observe t h a t if P is a point on ak, f

u ; (P) = u + ( P ) + J~k~ s = u~ (P) + tjk, (4) while for P on b~,

f a

u~ (P) = u; (P) +

j a

= u; (P) + rei~jk.

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Thus b y (1) above, we have t h a t on ak,

/- =/(u + + ~ eg+k) = exp [2 rd~g+k (Au + ~)]/+, (6) and d/- = exp [2 ~i~g+k (Au + ~)] (d/+ -~/+ 2rei~g+kAdu). (7)

Hence, 2x~i ^{k = l} ~ ~ ] + " : 2 ~ / k = l k k = l

A similar calculation, except t h a t now it is more convenient to express /+ in terms of / - , shows t h a t

2 ~ i k=l ~ k=l

Since a 1 b y 1 m a t r i x i s symmetric, ~kA~e~+k=~g+k~Ae~. Thus, we have

g g

k = l k = l

(N1 - h ~ 2 ) then we have simply I f we write the skew symmetric N as N2 N3

N(/) = trace N 2. (9)

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42 a. LEWITTES

Assuming again t h a t

/(u(P))~0

on 8, let P1 . . . P~(r) be the zeros of ! on S.

These are not necessarily distinct but each is repeated according to its multiplicity.

Again, by the residue theorem, we have, for a fixed index h, 1 ~< h ~<9, t h a t

N 1 ( d/

Y ~ ( P J ) = - - Jo ~ -/, 00)

.i=l 2 g i so

which is, in our previous notation,

1 + | | u h ~ - - u h 9

2~ik=~ k abk\ / Using (3) to (6), we obtain,

. = - -

thk ~ d! + UT- - thkeo+kAFtek-- . ~ ~ u+~g+kYAdu. h

2 Y~i j ak ! j a~

(11) B y assumption, !+ is different from zero along ak and a single-valued branch of, log !+ is defined in a neighborhood of ak. If Q~, Q~ are the (identified) initial and final points of a~, respectively, t h e n

fo~ d!+

~ - = log !+

(u(Qki)) -

log

!+ (u(Q'~)).

B u t !+

(u(Q~)) = !+ (u(Q~)

+ ~ ek) = exp [2

:~i ek (Au(Q~) +

y)] !+ (u(Q~)); so t h a t

fa d~!+ +- = 2 ui ~k (A u(Q~) + y) + 2 ~i vk, (12)

where vk is an integer, independent of h.

I n a similar way we find t h a t

1 fbk ( U ~ - - - - d]+ -d[-~ 1 ~ d ] - 2zeig~]kdu)- (u +~ --~ri~hk)all-

03)

Denote the initial and final end points of bh as Q~,

Q~,

respectively. The same argument which led to (12) gives,

d]- (14)

where gh is an integer.

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R I E M A N N S U R F A C E S A N D T H E T H E T A F U N C T I O N 4 3

Summing (11) and (13) over k from 1 to g and adding, taking into account (12) and (14), we have,

'~(f)

u h ( e j )

= - ~ t.~(ek(hu(Qg)+~.)+,~+~a+~Xnek) - ~ f a u+~.+kXduh

. i = l k = l k = l k

+ (15)

( n )

This formula is not very useful in its full generality. L e t us take A = O , - ~ i I , where O, I are, respectively, the g b y g zero and identity matrix, and n is a positive integer. L e t G, H E C g be arbitrary; take

~j=Gj, for l~<j~<g, and ~j= - ~ i ~ i t z - H s , n for g+l~<]~<2g.

(o o')

Then N = n and N ( / ) = n g . (15) now reduces to

N(f) Tt g

5 u.(Pj)= - ~ t.~(G~+~ + n ) + _ ~: ~ u ~ d u ~ - n - ( u . ( ~ ) + 8 9

j = l k - : l :7~'~ k = l , ) a ~

Define Ka ⁼ - - :Tg-i k = l k

and let (n), v, ~u, K, G, H, be g-rowed column vectors whose kth e n t r y is n, v~,juk, Kk, Gk, Hk, respectively. Thus the divisor ~(/)=/)1 ... Pg(I)satisfies

u(((/)) = T ( - (n) - a - v) - n K - xei H + =i # . (16) B u t - T ( n ) - T ~ + = i # = O , and (16) can be written as

u ( ( ( / ) ) =- - T G - n K - =i H . (16')

K is called the vector of Riemann constants. I t is independent of G and H but depends, along with u, on the base point B 0. This dependence will be clarified later.

We summarize our results in the following theorem.

T ~ O R ~ M 1. L e t /(u) be a n entire / u n c t i o n i n C ~ s a t i s / y i n g , /or l ~ k ~ < 2 g , / ( u + ~ek) = exp [2 ~ i ~ (/~u + ~)]/(u);

/(u) is a multiplicative / u n c t i o n on J ( S ) . B y m e a n s o/ the m a p u : S--> J ( S ) , / ( u ( P ) )

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4 4 J . LEWITTES

is a multiplicative /unction on S. Either /(u(P)) is identically zero on S or it has a divisor o/ iV(/) zeros, ~ = Px " " P N ( f ) .

~(/)=

trace 1V~ where

(

^~

I n the case that A = O , - z I = trary points in C g, the characteristic matrix

where H s - - -- 2~i t , - H j and G, H are arbi- ⁿ

N = (nO -- n~) and N ( / ) = n g .

~ (/) satis/ies the congruence u( $(/) ) + n K = - ~ ( H ) = - T G - ~ti H .

Functions having the particular form given above are called n t h order t h e t a functions, with characteristics G, H. Details concerning their construction and the n u m b e r of linearly independent ones, over the complex field, m a y be found in [1], pp. 91-104. The first order t h e t a function with characteristics G, H is uniquely determined up to a constant multiple. I t is

where

= exp ( ~ T G + 2 ~ u + 2zti ~ H ) 0 (u + T G + zdH),

0(u,:0 ( ou+o o)

m running over all g rowed column vectors with integer entries. A crucial p r o p e r t y of O(u) is t h a t it is even; O ( u ) = O ( - u ) , as is a p p a r e n t from its definition. I n fact, it is even in each variable separately.

Section II

I n this section we investigate closely only a particular first order 0 function;

namely, let e E C g be some given point and consider the function O ( u - e ) . This is the first order 0 with G = O , H = - (~ti) - l e . I n this case, Theorem 1 tells us t h a t either 0 ( u ( P ) - e) is identically zero on S, or it has a divisor of g zeros $, such t h a t e = u($) + K. Before proceeding we introduce some convenient notation. L e t S ~, n ~> 1, denote the cartesian product of S with itself n times and D ~ the symmetric product, i.e., the quotient space of S = under the symmetric group of permutations. Briefly, S ~

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R I E M A N N S U R F A C E S A N D T H E T H E T A F U I ~ C T I O N 45 consists of ordered n-tuples and D ~ of non-ordered n-tuples. D ~ is simply the set of integral divisors of degree n. A neighborhood of ~ = P 1 ...P~ E Dn shall m e a n the set of all divisors QI...Q~, where Qj belongs to some parametric disk on S about Ps- Recall t h a t u : D ~---> J(S). Denote b y W ~ ~ J(S) the image of D ~ : u(D ~) = W ~. We agree to set W ~ = 0 EJ(S) and D o= the unit divisor, 1-Ie~s p0. Since J ( S ) is an abelian group, sets of the form X + Y are defined. T h a t is, if X , Y c J ( S ) , X + Y denotes the set of elements of the form x § x E X , y E Y , and - X the set of elements of the form - x , x E X . I n particular, X + y is the set of elements of the form x + y , x E X .

THEOREm 2. O ( W g - I + K ) = O ; i.e., 0 ( u ) = 0 i/ u E W g - I + K .

Proo/. I t is well known t h a t there exists a divisor ~ E D g consisting of distinct points on S such t h a t i ( ~ ) = 0 , and t h a t if ~' belongs to a sufficiently small neighborhood of ~ it also has distinct points a n d i ( ~ ' ) = 0 . L e t ~ = P I ... Pa be such and set e = - u ( $ ) + K . Now, if O ( u ( P ) - e ) = O , t h e n for l~<]~<g,

0 = 0 (u(Pj) - e) = 0 (u(P~... l~j... Pg) + K),

where Pj denotes deletion of the point Pj. I n the last step we used the evenness of O(u). I f O ( u ( P ) - e ) S O , then b y Theorem 1 it has a divisor of g zeros w such t h a t e = u ( w ) + K. B y the construction of e we have t h e n u ( ~ ) = u(oJ), which inp]ies ~ = o~

(by our assumption t h a t i ( ~ ) = 0 a n d the R i e m a n n - R o c h and Abel theorems). Thus, again, we obtain 0 ( u ( P 1 . . . / ~ . . . Pg) + K) = 0. B y our r e m a r k s a t the s t a r t of the proof it is n o w clear t h a t O(u(Q 1 ... Q g - 1 ) § vanishes identically on a full neighborhood D g-l, hence is identically zero on D g-l, which completes the proof.

COROLLARY: O(Wr+ K ) = O , l <~r<~g-1 and 0 ( K ) = 0 . Thus, the vector o/ Rie- mann constants is always a zero o/ O(u).

Proo/. Although D r ~ : D t for t > r it is clear t h a t W r c W ~ ; for, if ~ E D ~, u(~)E W ~, t h e n B t o - ~ E D t and u(Bto-r~)~u(~), so u ( ~ ) E W t. This gives the first statement. To see t h a t O(K)= 0, we need only observe t h a t O~u(Bro)E W r, for every integer r.

THEOREM 3. Let ~, ~oED a and e E C a.

(a) I] O ( u ( P ) - e ) ~ 0 and has divisor o/ zeros ~, then i(~)= O.

(b) I / e =- u(o)) + K and O(u(P) - e) ~ O, then o~ is the divisor o/zeros.

(c) I / e =- u(eo) § K and i(o~) > O, then O(u(P) - e) =- O.

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46 j. LEWITTES

Proo/. (a) L e t Q E S be such t h a t O(u(Q)-e)~=O. I f i ( $ ) > 0 , there is a $ ' E D g s u c h t h a t Q appears in ~' a n d u(~') - u(~). T h e n b y T h e o r e m 1, e =- u(~') + K and, b y T h e o r e m 2, O ( u ( Q ) - e ) = 0, which contradicts our choice of Q.

(b) For, if $ is t h e divisor of zeros, e - : u ( ~ ) + K = u(r K. B u t , b y (a), i ( $ ) = 0, so t h a t ~ = w .

(c) I m m e d i a t e f r o m (a) a n d (b).

W e n o w wish to analyze t h e dependence of the m a p u a n d t h e R i e m a n n con- s t a n t s K on the base p o i n t B o. F o r clarification, if B0, B 1 . . . B ) . . . . is a set of points on S, we shall d e n o t e b y u ~ 1 .. . . , u j , . . . , a n d K ~ 1 .. . . . K j . . . t h e u a n d K ob- t a i n e d with respect to base p o i n t B o ,B 1 .. . . . Bj . . . Of course, e v e r y t h i n g we h a v e done up to this point has been t r u e for a n y choice of B0, hence we h a v e n o t w r i t t e n u ~ ~ until now. Also note t h a t t h e sets W T, r>~ 1, d e p e n d on t h e base point B0, a n d when necessary we m a y write W~o, W~,,..., W~j . . . O u r first result is

T H E 0 ~ M 4. (a) u l ( P ) : - u ~ 1 7 6 /or all P E S . (b) K l =- K ~ + u ~ (B~-l).

Proo/. (a) is trivial, for ~ , - - - ~ 0 - ~ ' , .

(b) Since O(u) is n o t identically zero in C g, there is an e E C ~ such t h a t 0 ( e ) 4 0 . T h e n O ( u l ( p ) - e ) ~ O , for P = B 1 is n o t a zero. L e t ~ be t h e divisor of zeros;

e ~ u l ( ~ ) + K 1. B y (a), we see t h a t O ( u ~ 1 7 6 has t h e same divisor of zeros

~; hence u ~ (B1) + e = u ~ (~) + K ~ C o m p a r i n g these t w o congruences for e gives (b).

A c t u a l l y (b) m a y be seen directly f r o m t h e definition of K. A n interesting consequence of this t h e o r e m is t h a t K 1 - - K ~ if a n d o n l y if u ~ 1 7 6 T h a t is, B o a n d B 1 a r e Weierstrass points on S such t h a t there is a f u n c t i o n with a pole of order g - 1 at B 0 a n d a zero of order g - 1 a t B 1. This is, in 'fact, t h e case for a n y two Weierstrass points on a hypereUiptic surface of o d d genus, b u t I do n o t k n o w w h e n else this occurs.

W e n o w investigate t h e dependence of t h e identical vanishing of O ( u ( P ) - e ) on t h e base point. As we h a v e a l r e a d y r e m a r k e d , if 0 ( e ) 4 0 , t h e n O ( u ( P ) - e ) ~ 0 for every base point; for P = base point, is n o t a zero. Thus, assume t h a t 0(e)= 0, B o, B 1 are t w o a r b i t r a r y base points, a n d O ( u J ( P ) - e ) ~ O for ~'=0, 1. L e t ~0, ~1 be t h e respective zero divisors. Clearly, B 0, B1, respectively, are zeros so t h a t ~o = Bo e~ ~1 = B1 o)1, where o)0, o ) I E D g-1. B y o u r previous results we h a v e :

e -~ u ~ (~o) + K ~ ~- u~ (~o) § K ~

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RIEMANN SURFACES AND THE THETA FUNCTION 47 a n d e ~2 u 1 ($i) + K1 = u~ (~ - u~ (B~ -I) + K0 + u0 (a~ -1) -- u~ ((91) + K~

Thus, u ~ (COo) -= u ~ (ool) , u ~ ($o) --- u~ (Bo ~ and, b y T h e o r e m 3, we infer t h a t ~-o = Bo ~ hence COo=CO r Since o o E D ~-1, there is a differential ~o with divisor eOoy , where y E D g-1. y is u n i q u e l y determined, for b y T h e o r e m 3, i(Bocoo)=0, which implies i(COo)=l. Now, let B 2 be a point of y a n d suppose O(u2(P)-e)~O. T h e n it has divisor of zeros ~2 which, b y o u r a b o v e a r g u m e n t (with B~ in place of B1) , m u s t be ~ = B ~ co 0 and, b y T h e o r e m 3, i(B2 COo)= 0. B u t ~v is, b y construction, a multiple of B 2 w 0, so t h a t i(Bz~%)=l, a contradiction. Thus, for each p o i n t B 2 of V, O(u ~ (P) - e) - O .

W e n o w show t h a t - - w i t h t h e same conditions on e, B 0 a n d B l - - t h e r e are a t m o s t g - 1 distinct points B E Y such t h a t O(uS(P)-e)=O (where u z means u with base p o i n t B). For, b y hypothesis, O(u~ e)4:0 for a n y p o i n t P0 E S n o t a p p e a r i n g in B 0 w 0. Since u~ we h a v e t h a t O(u~176 so t h a t , as a function of B E S, with P0 fixed, O(u ~ (Pc) - u~ (B) - e) ~ O. This has t h e n a divisor of zeros /3 satisfying u~176176 Now, if B 3 is n o t one of the g points of /3, we h a v e O(u ~ (P0) - u~ (B3) - e) 4: 0, i.e., O(u 3 (Pc) - e) 4= 0 a n d so O(u 3(P) - e) ~ O, as a function of P . Thus, if /3 contains fewer t h a n g distinct points, our assertion is proved.

Otherwise, let /5 = Q1 . . . Qg consist of g distinct points, such t h a t for each ~, 1 ~< ] < g, O(u Qj (P) - e) =- O. L e t /)1 E S, P I 4: P0, be a second p o i n t n o t in B 0 ~o o. Carrying t h r o u g h as before, we o b t a i n a congruence u ~ 1 7 6 ~ where 8 is t h e divisor of zeros of O ( u ~ 1 7 6 a n d O ( u S ( P ) - e ) , O , for e v e r y p o i n t B3 n o t in J.

I f o u r a s s u m p t i o n on /3 was correct, then, necessarily, /3 = ~, which implies u~

u~ which is absurd. Hence, /34:~, a n d o u r assertion is true. We s u m m a r i z e t h e above results as

T~I~OR]~M 5. Let eEC ~ satis/y O(e)=O. Either

(a) O ( u ~ /or every choice o/ base point B o E S , or ( b ) O(uQ~(P)-e)=--O /or at most g - 1 distinct points Q1 .. . . ,Qg-1.

I n case (b), there is uniquely determined a divisor o o E D g-l, i(O~o)= 1, such that i/

B E Y is not one o[ the points Q1 . . . Qg-1, then O ( u ~ ( P ) - e ) ~ O , and has divisoro/zeros

=Bco o. There is a uniquely determined di//erential q) o[ divisor eOo~ and the points o/

are included in the points Q1 .. . . . Qg-1.

Recalling t h a t t h e 0 f u n c t i o n is even, a n d b y (a) of T h e o r e m 4, we observe easily t h a t the following four s t a t e m e n t s are equivalent:

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4 8 J . L E W r r T E S

(1)

O(u~

for every base point

BoeS.

(2)

O(u~176

(as a function of P and Q) for every

BoeS.

(3)

O(u~176

for a particular

BoeS.

(4)

O(u~247

for every

BoeS.

Suppose t h a t (1) above holds. Differentiating

O(u~ (P ) - e) =- O

with respect to a local parameter at B o and setting P = B o gives,

~ 80 ( - e ) ~ j ( B 0 ) = 0

(~,-~du~).

Since b y hypothesis this holds for every point B ofi S, we see t h a t the differential

j=l ~uj ( - e ) ~ j

is identically zero. The linear independence of the ~j implies then

Ou--~j(-e)=O

80 for l < i < g .

On the other hand, suppose t h a t for a given

BoriS, O(u~

B y Theorem 5, we m a y assume B o different from the g - 1 points in w 0. Then

O(u~ e)~-0

has divisor of zeros Boc % and a simple zero at B0: Thus, the derivative of

O(u~

- - w i t h respect to a local parameter at B0--does not vanish at B 0. I n other words, j=l ~uj ( - e)~j(B~ 80

Thus we have proved

THEORElVi 6.

Let eeC g satis/y

0(e)=0.

Then O(u~ /or every base point B oeS i] and only i]

eu~ (-e)=O /~r 80 l < i < g .

The four equivalent statements given above and Theorem 6 show t h a t

80 ~0

- ~ j ( - e ) = 0 for l<~j~<g if and only if ~ u j ( e ) = 0 for l ~ < j 4 g . This fact, however, follows directly from the evenness of the 0 function.

Theorem 6 is a special case of the more general theorem of Riemann presented below. To prove the general theorem we shall follow the exposition o f Krazer, essentially, filling in certain gaps where necessary.

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RIEMAI~I~ S U R F A C E S A N D T H E T H E T A F U N C T I O N 49 So far, we h a v e shown t h a t if O ( u ~ for a base p o i n t B o E S , t h e n there is a divisor of zeros ~ E D g such t h a t e ~ u ~ ~ Thus, for points of t h e f o r m d = e - K ~ E C g we h a v e solved t h e J a e o b i inversion problem, i.e., f o u n d a divisor ~ E D g such t h a t u~ T h a t this can be done for a n y point d E C g i.e. u ~ Dg---~J(S) is o n t o - - c a n be p r o v e d quite easily w i t h o u t the 0 f u n c t i o n a p p a r a t u s . ([5], p. 284).

As we shall see though, the essential feature of R i e m a n n ' s solution via the 0 function, is t h a t one can describe precisely the pre-image in D g of a p o i n t e E C a b y means of the behavior of the 0 function at e.

L e t us first observe t h e following. O(Wg+K):~O, i.e. 0 does n o t vanish identically on t h e set W g + K c J ( S ) . (Note t h a t we h a v e suppressed m e n t i o n of t h e base p o i n t B 0 E S in t h e n o t a t i o n ; for, until f u r t h e r notice, we shall assume B o fixed a n d it will n o t be varied.) Indeed, if $ = P 1 . . . P g E D g consists of g distinct points t h e n i{~) = 0 is equivalent to the s t a t e m e n t d e t ( ~ ( P j ) ) 4 0 . This d e t e r m i n a n t is t h e J a - cobian of t h e m a p u: D g--> J(S), whose n o n - v a n i s h i n g a t ~ implies t h a t u is a local h o m e o m o r p h i s m . Thus, W g contains a n open set of J(S) a n d since O(u) is n o t identically zero on J(S), it c a n n o t vanish on W ~. Similarly, O(u) does n o t vanish identically on W g + e, for a n y e E C g.

Suppose n o w t h a t O(e)=O. T h e n there is a n integer s such t h a t O(W r - W r - e ) = 0 , for O~r<~s, while O ( W 3 - W S - e ) 4 0 . B y o u r previous r e m a r k s l ~ s ~ g - 1 . Thus, b y definition, there are co0, o o E D s such t h a t O(u(o)o)-u((xo))40. W e m a y assume t h a t ~og, o0 consist of 2s distinct points for, b y continuity, if Q a p p e a r s twice, we m a y m o v e one occurrence to a neighboring Q', still keeping t h e value of 0 a w a y f r o m zero. L e t co0 ~ P0 e~ ~~ EDS-1; t h e n O(u(P) + u(eoo) - u(oo) - e) ~ O, a n d has a divisor of zeros ~-0 E Dg. B u t , P = a point of o 0 is a zero, for t h e n u ( P ) + u ( e o ) - U ( O o ) - e is in W S - l - W S - l - e , where 0 vanishes. Thus, ~o=Oo/~o, /~oED a-~, a n d we h a v e t h e congruence -u(eoo)+U(Oo)+e=-U(Ooflo)+K , or e-=-u(eOoflo)+K, where ~00floED a-1. This proves the following extension of Theorem 2.

T~EORE~a 7. W~-I § K is the complete set o/ zeros o/ 0 on J(S).

Again, b y a c o n t i n u i t y a r g u m e n t , if toED s-1 is sufficiently near COo, O(u(P)+

u(co) - u(0"o) - e) ~ 0, a n d for a suitable fl E D ~-s, e ~ u(co/~) + K. To see the significance of this we m u s t pause for t w o lemmas.

L E P t a I. Let X be a topological space, F a /ield, /i . . . . /N /unctions on X to F.

~ 1 ~ /~, 2~ E F, ~ not all zero, vanishes Assume that no non.trivial linear combination/= N

on an open set o/ X . Then, given any N non-empty open sets V 1 .. . . . VN o/ X , there are points x~E V~ such that the determinant o/ (/~ (x~)) is not zero.

4 - - 6 4 2 9 4 5 A c t a mathematica. I I I . I m p r i m 4 le 12 m a r s 1964.

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5 0 g. LEWITTE$

This is a simple exercise in linear algebra a n d the proof m a y be omitted. I n particular we have the following corollary: I f X is a set, a n d ]1 . . . ]~ are functions on X to F which are linearly independent, t h e n there are points xl . . . xN E X such t h a t det (/~(x~))#O. This follows f r o m the l e m m a b y topologizing X indiseretely, only the e m p t y set a n d X are open, a n d taking all Vr = X.

L ~ M ~ A 2. Let tooED m, too=~'oc% ~oED ~, ~oED 'n-', O<~s<<.m. Suppose there is a neighborhood V o] ~o such that /or any ~EV there is a a E D m-" with

u(eoo)-u($a).

Then i(mo) >t g + s - m.

Proo/. Delete • from V a n d select a smaller open set V ' c V - $ 0 such t h a t E V' consists of distinct points which do not appear in %. Also, we m a y t a k e V' to be of the form V 1 • • Vs, where each Vj is a disk on S. B y hypothesis, for a n y

~ V ' , there is a a such t h a t u(~a)=-u(too). B y Abel's theorem, there is t h e n a function [ on S, with divisor ( f ) = ~a/to 0. B y our choice of V', ] has a t least s zeros a t

$, which are not cancelled b y a n y of the points of to 0. Now r(to$1) = m + i(too)+ 1 - g , and the space of functions which are multiples of too 1 has a basis of N + 1 linearly independent functions [~ . . . [N+I, where N = m + i ( t o o ) - g . B y L e m m a 1, if s > ~ N + l , we m a y select points PsEVj, I ~ < ~ < N + I such t h a t det (/~(Pj))=~0. This means t h a t no (non-trivial) linear combination, /, of the f u n c t i o n s / i . . . fN+I vanishes a t the points /)1 . . . P~+I. B u t P1 . . . P~+I m a y be completed to a divisor ~EV' and, as constructed above, there is a non-constant function vanishing a t P 1 - . . P~. This contradiction proves t h a t s < N + 1, f r o m which i ( w o ) > ~ g + s - m .

B y reversing the reasoning, one obtains a converse of the following form. I f i ( t o o ) > ~ g + s - m , t h e n for a n y ~ E D ~ there is a a E D m-s such t h a t u(too)=-u($a ). Let us call s the n u m b e r of free points of too, where s is the greatest integer such t h a t for ~ E D s there is a a E D m-s with u(too)==-u($a). B y the lemma, a n d the converse just stated, we h a v e the following.

COROLLARY: i(too) = g + S -- m, where too E D m and s is the number o] /tee points o] too.

I n particular, if m = g - n , 0 ~ < n ~ < g - 1 , t h e n i(too)=n+s.

Reverting to the discussion preceding the lemmas, we h a v e e ~ u ( t o o f l o ) + K and for to sufficiently near too, there is a fl such t h a t e = u ( t o f l ) + K , u(to0fl0)~=u(tofl). Thus, applying L e m m a 2, with s of t h e L e m m a as s - 1 a n d m as 9 - 1 , we h a v e i(toofl0)>~

g + ( s - 1 ) - ( g - 1 ) = s .

We h a v e shown t h e n t h a t O ( W r - W r - e ) = O , for O < r < . s - 1 , implies t h a t e - ~ u ( $ ) + K , where ~ E D g-1 a n d i($)>~s. Conversely, this latter s t a t e m e n t implies

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R I E M A N N S U R F A C E S A N D T H E T H E T A F U N C T I O N 51 O(W r - W r - e ) = O , for O < ~ r < ~ s - 1 . For, we are given i ( ~ ) > ~ g + ( s - 1 ) - ( g - 1 ) , so t h a t has a t least ( s - 1) free points, x E W ~ - 1 - W ~ - l - e is of t h e f o r m x = u ( P 1 ... Ps-~) - u ( Q 1 . . . Q ~ _ ~ ) - e , a n d we m a y write e = - - u ( P 1 . . . P s _ 1 8 ) + K , with O E D ~-~. This gives x - - u ( Q 1 . . , Q ~ _ x a ) - K , and, b y T h e o r e m 2, 0 ( x ) = 0 . Thus, O ( W ' - I - W ~ - I - e ) = O , and, since W ~ W s-l, for 0 ~ < r ~ < s - 1 , o u r assertion is proved.

W e n o w p r o v e t h a t O(W ~ - W ~ - e ) = 0 , for 0 ~ < r < s , implies t h a t all partial derivatives of 0 of order r vanish at - e (hence also a t e, for O(u) being even implies 0(W r - W ~ + e) = 0, for 0 ~< r < s). I n fact, more is true. N a m e l y , if 0( W ~-1 - W s-1 - e) = 0 t h e n

( r ) : ~ u j , . . . ~ u s ( W a - l - r - w ~ - ~ - ~ - e ) = O for O < ~ r < s - 1 a n d 1~<]1 . . . . ,]r<~g.

I f r = 0 we h a v e t h e 0 function itself. Since - e E W s - l - r - W S - l - r - e , we h a v e t h a t all partial derivatives of 0 of order up t o a n d including s - 1 vanish a t - e . N o w t h e s t a t e m e n t (r) a b o v e is t r u e for r = 0, b y hypothesis. Assume it has been p r o v e d for all r~<n, 0 ~ < n < s - 1 ; we shall prove it for n + l . W e h a v e

( u ( P 1 . . . P s - l - n ) - u ( Q 1 . . . Q s - l - n ) - e) = 0 ,

for a n y points P1 "" P , - 1 - n , Q1 ".. Q,-I-,~ on S. F i x p a r t i c u l a r choices for all of these p o i n t s except P1, which we let v a r y in a small n e i g h b o r h o o d of Q1- We h a v e t h e n a function of P1 which is identically zero, so t h a t differentiating with respect to a local p a r a m e t e r a t Q1 a n d setting P1 = Q1, we still h a v e zero. B y t h e chain rule f o r differentiation.

g ~-+10

t ~-1 ~Ut, . . . eOUt, DUi ( u ( P 2 "'" . P s - l - n ) - u ( O 2 ... Q , - 1 - n ) - e) d u j ( O l ) = O.

This differential is a linear c o m b i n a t i o n of t h e linearly i n d e p e n d e n t duj=qJj, w i t h coefficients i n d e p e n d e n t of Q1, which vanishes a t e v e r y p o i n t of S, as Q1 was arbi- t r a r y . Thus, each coefficient is zero which proves (r) for n + 1, completing t h e in- ductive proof.

W e come n o w t o t h e crucial point; n a m e l y t o show t h a t if O ( W ~ - W r - e ) = O , for 0 < r ~< s - 1, b u t 0(W s - W ~ - e) # 0, t h e n a t least one partial derivative of 0 of order s does n o t vanish a t - e . As we h a v e a l r e a d y observed, O ( W ~ - W ~ - e ) # O implies there are divisors of 28 distinct points, w0, a o E D s such t h a t 0(u(eo0) - u(a0) :- e) # 0.

B y continuity, for ~ q D ~ sufficiently near a 0, O(u(wo) - u(z) - e) :4: O. Also, O(u(v) -

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5 2 J . L E W I T T E S

u(oo)- e)

c a n n o t v a n i s h for all T n e a r a0, for, otherwise, t h i s f u n c t i o n of s v a r i a b l e s w o u l d v a n i s h on a n o p e n set, a n d so w o u l d b e i d e n t i c a l l y zero, c o n t r a d i c t i n g t h e e x i s t e n c e of oJ o. Thus, t h e r e is a T0 6 D s such t h a t

O(U(To)- U(Oo)- e):4=0, O(u(~Oo)-

u(T0) -- e) ~: 0, a n d a g a i n , b y c o n t i n u i t y , T 0 m a y b e a s s u m e d t o h a v e d i s t i n c t p o i n t s , a l l d i f f e r e n t f r o m t h o s e in w o a n d 0 0.

L e t dq d e n o t e t h e n o r m a l i z e d a b e l i a n d i f f e r e n t i a l of t h i r d k i n d o n S , w i t h zero p e r i o d s o n a ~ , . . . , a o, a n d w i t h r e s i d u e + 1 a t t h e p o i n t s of ~o a n d - 1 a t t h e p o i n t s .of 0 0 . T h u s , ff o 0 = Q t ... Q~, T 0 = R 1

... R~, d ~ = ~ = t d~?o~.Rj,

w h e r e d~oj.a ~ is t h e a b e l i a n d i f f e r e n t i a l of t h i r d k i n d o n S w i t h zero p e r i o d s o n a~ . . . . ,a~ a n d r e s i d u e + 1 .at Rj a n d - 1 a t Q~. R e c a l l t h a t t h e R i e m a n n p e r i o d r e l a t i o n s for s u c h d i f f e r e n t i a l s a r e

fb d~oi. ~ = 2 (uk (Rj) - uk

(Q~)) (1 ~< k ~< g, 1 ~< j -<. s), (17)

k

nvhere

uk(R,)--uk(Q,)=~q~k

is t a k e n o v e r a p a t h f r o m Qj t o Rj l y i n g c o m p l e t e l y in t h e s i m p l y c o n n e c t e d r e g i o n S o . W e a r e a s s u m i n g here, as is c l e a r l y p e r m i s s i b l e , t h a t t h e p o i n t s of 0 o, T0 lie on S o. Consider n o w t h e following f u n c t i o n of s p o i n t s on S,

O ( u ( P l " " P s ) - u ( a ~ ( ~=~f;i ) ](P1 ... P~)=O(u(p1 p~)_u(.ro)_e).E,

w h e r e E = e x p J d~) .

] is n o t a l w a y s zero o v e r zero, for a t P a . . - P s = too i t h a s a f i n i t e v a l u e . C o n s i d e r f o r t h e m o m e n t / ) 2 - . - P ~ fixed, a n d e x a m i n e ] a s a f u n c t i o n of />1. F o r P~ ... P , f i x e d a t v a l u e s s u c h t h a t n u m e r a t o r a n d d e n o m i n a t o r do n o t v a n i s h i d e n t i c a l l y i n P1, ] is a single v a l u e d m e r o m o r p h i c f u n c t i o n of P1 o n all S. T h i s follows d i r e c t l y f r o m t h e p e r i o d p r o p e r t i e s of 0 a n d t h e r e l a t i o n s (17). W e c l a i m n o w t h a t t h i s func- t i o n of P1 is a c o n s t a n t . I n d e e d , l e a v i n g a s i d e t h e q u a n t i t y E for t h e p r e s e n t , / ( P j ) h a s zeros d u e t o t h e zeros of 0 i n t h e n u m e r a t o r , w h i c h a r e g i n n u m b e r . B y o u r i h y p o t h e s i s t h a t

O(W r -

W r - e ) = 0 , for 0 < ~ r < s , s of t h e s e zeros a r e a t a0. T h u s , t h e 9 d i v i s o r of zeros for t h e n u m e r a t o r is o0~ , ~,ED g-8, a n d t h e r e is a c o n g r u e n c e ,

- u(P~

... Ps)

⁺U(Oo) + e ~ u ( ~ o Z) + K , o r e ~ u ( P , ... P~ ~) + K .

B y T h e o r e m 3, t h e d i v i s o r of zeros (roy has i ( o 0 y ) = 0 . I n t h e s a m e w a y , t h e d i v i s o r o f zeros for t h e d e n o m i n a t o r is ToS, 5ED~-~; w e a g a i n h a v e a c o n g r u e n c e

e -~ u(P~ ... Ps 5) + K,

a n d i(~o 5) = 0.

~rhus, u ( y ) - - u((~). I f ~ =~ 5, t h e n , b y t h e R i e m a n n - R o c h a n d A b e l t h e o r e m s , i(~) >~ s + 1.

" t h e R i e m a n n - R o c h t h e o r e m also shows t h a t a d d i n g a p o i n t t o a d i v i s o r d e c r e a s e s

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R I E M A N N S U R F A C E S A N D T H E T H E T A F U N C T I O N 53 t h e index i b y at m o s t one. Thus, i(ao7 ) >~ 1, which contradicts the fact t h a t i(o0~ ) = 0.

Thus, ~ = ~ , a n d t h e o n l y zeros a n d poles of /(P1) due to t h e q u o t i e n t of O's, are t h e zeros at 00 a n d t h e poles at z0. Consider n o w E, which, as a f u n c t i o n of P1, contributes the t e r m exp (~~ PI This is finite, a n d n o t zero, for P1 n o t one of t h e points in 00 , T0. As P I varies in a n e i g h b o r h o o d of a point Q of o 0 with local

(]Bod~) is, up to a finite non-zero factor, p a r a m e t e r z, z(Q)= O, exp P

e x p / / ' { | P ' = z l - l dz}\ = e x p ( - log z l + log z0).

\ J P0=zo Z /

Here % = z(Po) is a r b i t r a r y , as long as z0=~0. L e t t i n g z I -->0, we see t h a t E(P1), as P1--> Q of 0 o, has a simple pole. This cancels with the zero a t Q in o o f r o m t h e 0 in the n u m e r a t o r . On t h e other hand, using a similar n o t a t i o n at a p o i n t R of T o, we see t h a t , as P1--->R, E(P1) behaves like exp (SZz~=elz-ldz) as zi-->0. T h a t is, E(P1) at P1 = R has a simple zero, which cancels the pole due t o t h e zero of t h e d e n o m i n a t o r at P1 = R. Thus, all zeros a n d poles cancel, a n d /(1)1) m u s t be a con- s t a n t C.

T h e c o n s t a n t C depends on P2 -.. P,. B u t , / is s y m m e t r i c in P1 .-- P,, its value is n o t d e p e n d e n t on the order of t h e points P1 ... P~. This implies t h a t if [ is con- s t a n t in Pz, t h e n it is a c o n s t a n t in all s variables.

W e h a v e t h e n t h e following equation:

CO(u(P 1 ... P~) - u(v0) - e) = O(u(P 1 ... Ps) - U(Oo) - e) E. (18) Differentiate (18) with respect t o (a local p a r a m e t e r z at) P1 a n d set P1 = R r This yields

C Z ao

]~=1

~Ui

(u(Pz "" Ps) - u(R2 ... Rs) - e) dur (Rj) L ~0

= .t~=1 ~U 1

(u(R1Pz'." Ps) - U(ao) - e) du~ (R1) E (R1)

+ O(u(R 1P~ ... P~) - u(ao) - e) d E

(R1), (19)

where

o ~ J B, /

As we h a v e a l r e a d y r e m a r k e d , E(R1) has a simple zero, so t h a t t h e first t e r m on t h e right is zero. dE(R1) is a finite non-zero q u a n t i t y , essentially of t h e f o r m exp ( ~ = 2 S~okd~).

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54 J . L E ~ / I T T E S

Thus, if we now differentiate (19) with respect to P2, and set P 2 = R 2 , we have g ~2 0

C ~ (u(P 3 ... P~) - u ( R 8 ... R~) - e) duy, (R1) duj, (R2)

11,1~=1 ~UYx ~U)2

= j~-I ~ (u(R, R~ Pa "" P~) - u(ao) - e) guy (R2) (dE(R1)) (R2)

+ O(u(R1 R2 Pa ... P~) - U(~o) - e) d (dE(R1) (R2).

Again, the first t e r m on the right vanishes, for it is essentially exp (SR'd~y) which has a simple zero a t R~, while the second t e r m is a non-zero quantity, essentially of the form exp ( ~ = 3 S~d~y). Continuing in this fashion, we finally obtain, after differentiating s times,

C ~ asO

.i ... i,=1 ~UJ, . . . ~Uj, ( - e) duy 1 (R1)... duy, (Rs) = 0 (u(R1... Rs) - U(ao) - e) F , (20) where F is a finite non-zero q u a n t i t y due to the factor E. B u t our construction assured from the outset t h a t O(u(vo)- u((7o)- e)~=0. Thus, not all coefficients on the left of (20) vanish, so t h a t some partial of 0 of order s does not vanish a t - e . We collect our results in the following m a i n theorem.

THEOREM 8. Let e E C g. I / 0 ( e ) 4 0 , then e ~ u ( ~ ) + K /or a unique ~ E D g, and i(~)=O. I / O(e)=O, let s, l ~ < s ~ < g - 1 , be the least integer such that O ( W s - W S - e ) : # O . T h e n there is a ~ E D g-l, i ( ~ ) = s , such that e - u ( ~ ) + K . All partial derivatives o/ 0 o/

order less than s vanish at e while at least one partial derivative o/ order s does not vanish at e. The integer s is the same /or both e and - e .

Observe t h a t there is no mention a t all here of the base point B 0. This is to be expected, for the order of vanishing of 0 a t a point in C g is independent of the choice of B 0. Indeed, b y (a) of Theorem 4, a set of the form W r - W r is uniquely determined in J ( S ) , independently of the point B 0, even though W r is not. Thus it is only in considering unsymmetrie expressions of the form O ( u ( P ) - e ) , i.e., O ( W l - e ) , t h a t the base point B o plays a role.

One other point needs clarification. When it is stated t h a t e ~ - - u ( $ ) + K for some

~ E D a-1 with i ( ~ ) : s , t h e n this implies t h a t if also e - - u ( w ) + K for w E D g-l, then i(~o) = s . F o r completeness this is stated as a lemma.

LEMMA 3. Le ~, o~ED n, and suppose u ( ~ ) = u ( w ) . Then i(~)=i(o)).

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R I E M A I ~ I ~ S U R F A C E S A N D T H E T H E T A F U N C T I O N 55 Proo/. The hypothesis implies, b y Abel's theorem, t h a t there is a function /~

with divisor ~//(9. E x t e n d to a basis /1,/~I...,/N of t h e space of functions which are multiples of oJ. B y t h e R i e m a n n - R o c h theorem, N = n § i((9) + 1 - g. T h e functions h I = 1,

h~ =/2//1

. . . h~ =/N//1, are linearly i n d e p e n d e n t multiples of ~, so t h a t N ~< n § i(~) + 1 - g . Thus, i(w)~< i(~), a n d interchanging ~ a n d eo gives t h e result:

Section IlI

T h e integer s of T h e o r e m 8 a c t u a l l y has 8 9 as a n u p p e r bound. For, if e ~ u ( ~ ) + K , ~GD a-1 a n d i($)=s, choose t h e s - 1 free points of ~ at Bo, t h e base p o i n t of u. W e m a y assume t h a t B o is n o t a Weierstrass point, as T h e o r e m 8 is i n d e p e n d e n t of t h e base point. Thus, e ~ u ( ~ o -1 09) + K, where (9 ~ D g-~. Now, i(B~ -1) = g - (s - 1), the n u m b e r of gaps greater t h a n s - 1 at B0, a n d certainly then, i(~0 -1 (9) g - ( s - l ) . B u t i ( B ] - l ( 9 ) = i ( $ ) = s , so t h a t s < ~ g - ( s - 1 ) , or s ~ < 8 9 Combining this fact with T h e o r o m 8 yields t h e following:

THEOR]~M 9. Let T be a g• matrix o~ complex numbers, symmetric, with negative de/inite real part, and O(u) the associated 0 /unction /or T. Then i/ 0 has order > 1 (g + 1) at some point e E C a, i.e., 0 and all partial derivatives o/ order <~ 1 (g + 1) vanish at e, then T is not (the second hall o / ) a normalized period matrix o/ a Riemann sur/ace o/ genus g.

W e r e t u r n n o w to t h e considerations of the first p a r t of Section I I to consider once again t h e role of t h e base p o i n t B 0. Therefore, we write u ~ K ~ etc., as before.

Assume O(u~ Let s be t h e least integer such t h a t O ( W S + l - W ~ - e ) 4 0 ; here, of course, s depends on B 0. Then, there are 00, v o E D ~, which we m a y assume to consist of distinct points, such t h a t O(u ~ (P) + u ~ (00) - u ~ (v0) - e) ~ 0. This has a divisor of g zeros, s of which are at t h e points of T 0, so t h a t $=vofl o,floED a-~, a n d e - u ~ (a0flo) § K~ The same holds for q sufficiently near a0. F o r e v e r y such a there is a fl such t h a t e - u ~ 2 4 7 ~ Thus, aofl o has at least s free points, a n d b y L e m m a 2 then, i(aoflo)~s. This shows t h a t O(u~ implies a congruence e =- u ~ (~) + K ~ $ E D g, a n d i (~) ~> s >~ 1. W e can n o w complete T h e o r e m 3 to cover all eases b y adding t h e following: if e ~- u ~ (~) + K ~ $ e D a a n d i($) = 0, t h e n O(u ~ (P) - e) ~ O.

Also, it is n o w clear t h a t in t h e second case of T h e o r e m 5 the g - 1 points, at most, for which O(uJ(P)- e ) ~ 0 , are precisely t h e points of ~.

T h e a b o v e results enable us t o o b t a i n a characterization of the Weierstrass p o i n t s on S in terms of t h e 0 function a n d t h e R i e m a n n constants.

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5 6 j . LEWITTES

THe.OR~.M 10. Let B o E S be arbitrary, u ~ K ~ the m a p u and R i e m a n n constants K with base point B o. Then, B o is a Weierstrass point i / a n d only i / O ( u ~ ( P ) - K ~ - 0 .

Proo[. K ~ = u ~ (Bg) + K ~ B y our preceding remarks, O(u ~ (P) - K ~ - 0 if a n d only if i(Bg)>/1, which is the condition for a Weierstrass point. Note, t h a t if B 0 is a Weierstrass point, we n e e d n ' t have O(u 1 ( P ) - K ~ - 0 for every base point B r This, in fact, b y Theorem 6 (or Theorem 8), occurs if and only if

~0

~uj - - ( K ~ for l~<j~<g.

B y Theorem 8, this is if a n d only if K ~ ~ K ~ for ~ E D a-1 and i(~)/> 2. B u t K ~ 1 7 6 1 7 6 and i(B~-1)~>2 if and only if there are two gaps greater t h a n g - l, which is not true for every Weierstrass point. F o r example, on a "general"

surface with g ( g ~ - 1 ) Weierstrass points, a t which the gaps are 1,2 . . . g - l , g + 1, we have i(B~ -1) = 1.

L e t us now prove the following classical result.

THEORe.M 11. For a n y B o E S , i/ A E D 2a-u, then u ~ - - 2 K ~ i/ and only i/

A ks the divisor o/ zeros o[ a di[/erential on S.

Proo/. L e t ~ E D a-1 be a r b i t r a r y a n d set e = u ~ ~ Then, b y Theorem 7, 0 ( e ) = 0 and 0 ( - e ) = 0 ( e ) = 0 . Thus, for s o m e

~'ED g-i, -e~--u~ ~

Adding, we h a v e - 2 K ~ 1 7 6 where ~ $ ' E D 2a-2. Since ~ ' has g - 1 free points, we have i ( ~ ' ) = g + ( g - 1) - ( 2 g - 2) = 1, a n d $~' is the divisor of a differential. I f u ~ (A)-- - 2 K ~ t h e n u ~ 1 7 6 ') and the theorem follows from L e m m a 3, i ( ~ ' ) = i ( A ) .

The 0 function enables one, in certain cases, to obtain explicitly the linear combination of the normalized differentials ~01 .. . . . ~a, which vanishes at given points. F o r example, let ~ E D a-1 satisfy i ( $ ) = 1; there is t h e n a uniquely determined ~' E D a-1 for which i($$') = 1. Set e - u ~ ($) + K ~ which, b y Theorem 4, determines e E C a (modulo

~ , of course) independently of B 0. Consider

~0= L - ao

(-e)~s.

I=I ~U)

B y Theorem 8, since i(~)= 1, s for e is 1, and v 2 is n o t the zero differential. L e t B 1

be a point in ~, e - u I ( ~ ) + K 1 - u I ( B I ~ ) + K 1. I f i ( B 1 ~ ) = 1 , then, b y Theorem 3, O ( u l ( P ) - e ) - O . Differentiating and setting P = B 1 gives

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RIEMANN SURFACES AND THE THETA FUNCTION 57 y,(BO = ~ 80

j = l ~ U / ( - e ) d u j ( B 1 ) = O '

so t h a t v 2 vanishes a t B I. I f i(B 1 ~)= O, then O(u 1 ( P ) - e) has its g zeros at B a ~. As B 1 is in ~, it is a double zero, a n d again b y differentiating we h a v e

~(B1)=

0. Since u ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 ~ and, b y the same argument,

~

⁰

j=l (e) j vanishes a t $'. But, b y the evenness of 0,

eo (e) = v,'

Ouj Ou~ ( - e), a n d = - ~ .

Thus, yJ vanishes a t the points of ~$'. Also, if B is not in ~$', then yJ(B):t:0. For, e =- u B (B$) + K B, i(B$) = 0, and O(u B (P) - e) has a simple zero only a t B, hence ~(B) ~: 0.

Note t h a t we have not proved t h a t ~ has $$' as its divisor of zeros; b u t only t h a t y~ has a zero a t each of the distinct points of ~ ' . However, we have proved the following particular case:

THEOREM 12. Let A E D ~g-u be the divisor o/ a differential y~. Assume that A contains a divisor ~ o/ g - 1 distinct points, satis/ying i(~)= 1. Then, up to a constant multiple,

80

where e=u~ + K ~ /or any base point B ~

A point e E C o which has the p r o p e r t y 2e ~ 0 m a y be called a half period. Any half period is necessarily of the form e=Tde'/2 + T e l 2 where s, e' are integral vectors in C ~ Modulo ~ , there are 22a distinct haft periods, obtained b y letting the entries in e and e' be 0 or 1 in all possible ways. R i e m a n n calls a half period even if g e ' ~ 0 (mod 2), odd if ge '= 1 (rood 2). An easy calculation shows t h a t there are 2g-1(2 ~ 1) odd a n d 2 ~ 1 7 6 even half periods, (see [4], p. 8 of the Supplement). The mo- tivation for this even-odd terminology is the following. Recall the definition of 0 ] G ] ( u ) g i v e n a t the end of Section I; for the h a l f p e r i o d e = x d e ' / ' 2 + Tel2 consider

| e ( 2 | ( u ) . Then this function is an even or odd function of u according the function 0 [ s / 2 J

](0) and 0( ie'/e + Te/21

a s i e ' ~ 0 or 1 (mod 2); see [1], p. 103-4. Since 0 [e//2J

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5 8 J. LEWITTES

fer b y a n o n zero, exponential, factor, t h e order of t h e 0 f u n c t i o n a t a half period is the s a m e as t h e order of t h e corresponding 0 w i t h characteristics a t t h e origin.

W e recall t h a t a n o d d f u n c t i o n a l w a y s vanishes a t t h e origin a n d t h a t a p a r t i a l de- r i v a t i v e of a n e v e n (odd) f u n c t i o n is o d d (even). I n particular, a n o d d f u n c t i o n m u s t h a v e o d d order a t t h e origin; if all p a r t i a l d e r i v a t i v e s of o r d e r < s v a n i s h a t t h e origin, while s o m e p a r t i a l d e r i v a t i v e of order s does not, t h e n s is odd. A n e v e n f u n c t i o n has e v e n order a t t h e origin.

L e t e r . . .

e CN),

N = 2 g - l ( 2 g - 1), be a n e n u m e r a t i o n of t h e o d d half periods. B y o u r a b o v e r e m a r k s , t h e 0 f u n c t i o n vanishes a t each e r B y T h e o r e m 8, eZ)~ u(~ r + K , where ~(J) E D g-l, i(~ r = s s > / l , a n d sj is odd. I n particular, 0 - - 2e Cj) ~ u(~ ~i) ~(J)) + 2 K , and, b y T h e o r e m 11, t h e r e is a differential ~r w i t h divisor (~r Such differentials - - o r a c t u a l l y square roots of t h e m - - R i e m a n n , [4] p. 488, called abelian functions.

On t h e o t h e r h a n d ,

O(u)

need n o t v a n i s h a t a n e v e n half period. I f this occurs, it m e a n s t h a t t h e surface S h a s some special p r o p e r t y . F o r e x a m p l e , R i e m a n n ([4], p. 54 of t h e S u p p l e m e n t ) s t a t e s t h a t for g = 3, S is hyperelliptic if a n d o n l y if 0 vanishes a t some e v e n half period. T o see this, let us suppose first g a r b i t r a r y a n d e a n e v e n half period such t h a t 0 ( e ) = 0 . B y T h e o r e m 8,

e - u ( ~ ) + K,

i ( ~ ) = s ~ > l ,

~ED a-1.

As s m u s t be even, s >~2. Since 0 ~ u ( ~ 2) + 2K, t h e r e is a differential ~0 w i t h divisor

~2 ED2a-2

a n d since i(~)i> 2, t h e r e is a second differential ~0, w i t h divisor ~w, ~o E

D a-l,

~ = c o . ~1/~o is a f u n c t i o n w i t h divisor

~/oo;

b y A b e l ' s t h e o r e m , u ( ~ ) - u ( o ~ ) . Hence,

e~u(oo) +K,

which implies t h a t t h e r e is a differential ~ w i t h divisor eo 2. Clearly, the differentials ~0, ~o, ~ are linearly i n d e p e n d e n t , while t h e q u a d r a t i c differentials ~2 a n d

~0~ b o t h h a v e t h e s a m e divisor of zeros ~2 co 2. Thus, ~0 ~ = ~t(~0~), for some c o n s t a n t ;t.

W e see t h a t 0 vanishing a t a n e v e n half period leads to a linear relation b e t w e e n q u a d r a t i c differentials which are p r o d u c t s of (abelian)differentials. Suppose n o w g = 3 a n d 0 ( e ) = 0 , e a n e v e n half period. T h e n a relation of t h e f o r m ~o2=2(~0~), where

~o, ~, ~ are linearly i n d e p e n d e n t , b y a well-known result called ~ o e t h e r ' s t h e o r e m , implies S is hyperelliptic. H o w e v e r , we do n o t h a v e t o a p p e a l to N o e t h e r ' s t h e o r e m . S i m p l y o b s e r v e t h a t /=~0/~o is a f u n c t i o n w i t h divisor ~/o); as ~o has o n l y t w o p o i n t s w h e n g = 3, / is a f u n c t i o n w i t h t w o poles on S, a n d S is hyperelliptic. T h e converse, t h a t S hyperelliptie a n d g = 3 i m p l y 0 ( e ) = 0 for a n e v e n half period, will follow below f r o m our general discussion of hyperelliptic surfaces.

L e t S b e hyperelliptic a n d B o ~ S a W e i e r s t r a s s point. Since 2 g - 1 is a g a p a t Bo, t h e r e is a differential on S h a v i n g all its zeros a t Bo, i.e., h a v i n g divisor B~ ~-e.

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RIEMANN" S U R F A C E S AN'D T H E T I t E T A FU'N'CTION 59 B y Theorem 11, • 1 7 6 1 7 6 1 7 6 and so K ~ is a half period. B y our remarks after Theorem 4, we see t h a t if g is odd, the 2g + 2 Weierstrass points on S give rise to only one half period K ~ while if g is even, there are 2g + 2 distinct half period vectors K ~ Is K ~ even or odd half period? This is answered b y

THEOREM 13. Let S hyperelliptic with genus g = 4 k + m , /c~>0, 0 ~ m ~ < 3 . Let K ~ be the vector o/ Riemann constants with respect to a Weierstrass point B o E S. Then K ~ is a hal/period, even i/ m = O or m = 3 , and odd i/ m = l or m = 2 . W h e n m = l or2, the 0 /unction has order 2k § 1 at K ~ while when m = 0 it has order 2k and when m = 3 it has order 2 k § at K ~

Proo]. K ~ 1 7 6 1 7 6 and, b y Theorem 8, the order of 0 at K ~ is i(B~-l).

B u t

i(B~ -1)

is the number of gaps greater than g - 1 at B0, which is the number of odd numbers in the sequence g, g + 1 . . . 2g - 1. When m = 0, g = 4k, the gaps are g + 1, g + 3 . . . 2 g - l , and i ( B ~ l ) = 8 9 which is even. Since O(u) has even order at K ~ K ~ is an even half period. Similar considerations for the cases m = 1, 2 or 3 give the rest of the theorem.

I n the hyperelliptic case we m a y catalogue all even and odd half periods which zeros of O(u) in the following way. Let A 1 .. . . . A2g+2 be the 2 g + 2 Weierstrass points on the hyperelliptie surface S. Consider all divisors of degree g - 1 of the form:

~.<s) = A~ n A], ... Aj~_,_~ n' (21)

where 0 ~ < 2 n ~ g - 1 , l~<]k~<2g+2, l ~ < k ~ < g - l - 2 n , and jk4im if /c~:m. We have already seen t h a t a n y half period e, such t h a t 0(e)= 0, gives a divisor ~ of degree g - 1 with e-= u($) + K, and ~2 is the divisor ~ of a differential. Let us call such a a half period divisor. We now prove the following

THV.OR~M 14.

(a) Every hal/period divisor ~ is equivalent to a divisor ~n.(j) o/ the /orm (21).

(b) i(~.(j)) = n + 1.

(e) / / ~n.j):4:~n..r then these divisors are also not equivalent.

Proo/. (a) I t is well known t h a t on a n y hyperelliptic S there is a unique in- volution (automorphism of order 2) which leaves the 2g + 2 Weierstrass points fixed.

Also, if P is the image of the point P, not a Weierstrass point, under this involu- tion, then the order of a n y differential a t P equals the order of t h a t differential a t P , and P P , . , A ~ . Since $2 is the divisor of a differential, for every appearance of P,